Advanced wireless networks 4g technologies phần 5 pps

JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF 331 The complementary distribution function F C T H (t)is F C T H ( t ) = 1 − F T H ( t ) = F T H ( t ) = ⎧ ⎨ ⎩ 1 − e −μ M t − e −μ M t 1 + γ c  F T n ( t ) + γ c F T h ( t )  , for t ≥ 0 0, elsewhere (11.5) By differentiating Equation (11.4) we get the probability function (PDF) of T H as f T H ( t ) = μ M e −μ M t + e −μ M t 1 + γ c  f T n ( t ) + γ c f T h ( t )  − μ M e −μ M t 1 + γ c  F T n ( t ) + γ c F T h ( t )  (11.6) To simplify the analysis the distribution of T H is approximated in References [75, 76] by a negative exponential distribution with mean ¯ T H (1/μ H ). From the family of negative exponential distribution functions, a function which best fits the distribution of T H ,by comparing F C T H ( t ) and e −μ H t is chosen which is defined as μ H ⇒ min μ H ∞  0  F C T H ( t ) − e −μ H t  dt (11.7) Because a negative exponential distribution function is determined by its mean value, we choose ¯ T H (1/μ H ), which satisfies the above condition. The ‘goodness of fit’ for this approximation is measured by G =  ∞ 0   F C T H ( t ) − e −μ H t   dt 2  ∞ 0 F C T H ( t ) dt (11.8) In the sequel the following definitions will be used: (1) The probability that a new call does not enter service because of unavailability of channels is called the blocking probability, P B . (2) The probability that a call is ultimately forced into termination (though not blocked) is P F . This represents the average fraction of new calls which are not blocked but which are eventually uncompleted. (3) P fh is the probability that a given handoff attempt fails. It represents the average fraction of handoff attempts that are unsuccessful. (4) The probability P N that a new call that is not blocked will require at least one handoff before completion because of the mobile crossing the cell boundary is P N = Pr { T M > T n } = ∞  0  1 − F T M ( t )  f T n ( t ) dt = ∞  0 e −μ M t f T n ( t ) dt (11.9) JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 332 MOBILITY MANAGEMENT (5) The probability P H that a call that has already been handed off successfully will require another handoff before completion is P H = Pr { T M > T h } = ∞  0  1 − F T M ( t )  f T h ( t ) dt = ∞  0 e −μ M t f T h ( t ) dt (11.10) Let the integer random variable K be the number of times that a nonblocked call is success- fully handed off during its lifetime. The event that a mobile moves out of the mobile service area during the call will be ignored since the whole service area is much larger than the cell size. A nonblocked call will have exactly K successful handoffs if all of the following events occur: (1) It is not completed in the cell in which it was first originated. (2) It succeeds in the first handoff attempt. (3) It requires and succeeds in k −1 additional handoffs. (4) It is either completed before needing the next handoff or it is not completed but fails on the (k +1)st handoff attempt. The probability function for K is therefore given by Pr{K = 0}=(1 − P N ) + P N P fh Pr{K = k}=P N (1 − P fh )(1 − P H + P H P fh ){P H (1 − P fh )} k−1 , k = 1, 2, (11.11) and the mean value of K is ¯ K = ∞  k=0 k Pr{K = k}= P N (1 − P fh ) 1 − P H (1 − P fh ) (11.12) If the entire service area has M cells, the total average new call attempt rate which is not blocked is M Rc , and the total average handoff call attempt rate is ¯ KM Rc . If these traffic components are equally distributed among cells, we have γ c = ( ¯ KM Rc )/(M Rc ) ≡ ¯ K. 11.2.1 Channel assignment priority schemes The probability of forced termination can be decreased by giving priority (for channels) to handoffattempts(overnewcallattempts). Inthissection,twopriorityschemesare described, and the expressions for P B and P fh are derived. A subset of the channels allocated to a cell is to be exclusively used for handoff calls in both priority schemes. In the first priority scheme, a handoff call is terminated if no channel is immediately available in the target cell (channel reservation – CR handoffs). In the second priority scheme, the handoff call attempt is held in a queue until either a channel becomes available for it, or the received signal power level becomes lower than the receiver threshold level (channel reservation with queueing – CRQ handoffs). 11.2.2 Channel reservation – CR handoffs Priority is given to handoff attempts by assigning C h channels exclusively for handoff calls among the C channels in a cell. The remaining C − C h channels are shared by both new JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF 333 0 E μH 2μ H (C-C H )μ Η (C-C H +1)μ Η 1 E C-Ch E C-Ch+1 E C E E 0 E 1 E C-C h E C-C h +1 C E Λ R + Λ Rh Λ R + Λ Rh Λ R + Λ Rh Λ Rh Λ Rh Λ Rh Cμ H Figure 11.21 State-transition diagram for channel reservation – CR handoffs. calls and handoff calls. A new call is blocked if the number of available channels in the cell is less than or equal to C h when the call is originated. A handoff attempt is unsuccessful if no channel is available in the target cell. We assume that both new and handoff call attempts are generated according to a Poisson point process with mean rates per cell of  R and  Rh , respectively. As discussed previously, the channel holding time T H in a cell is approximated to have an exponential distribution with mean ¯ T H (1/μ H ). We define the state E j of a cell such that a total of j calls is in the progress for the base station of that cell. Let P j represent the steady-state probability that the base station is in state E j ; the probabilities can be determined in the usual way for birth-death processes discussed in Chapter 6. The pertinent state-transition diagram is shown in Figure 11.21. The state equations are P j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  R +  Rh jμ H P j−1 , for j = 1, 2, ,C −C h  Rh jμ H P j−1 , for j = C −C h + 1, ,C (11.13) As in Chapter 6, by using Equation (11.13) recursively, along with the normalization con- dition ∞  j=0 P j = 1, the probability distribution {P j } is P 0 =  C−C h  k=0 (  R +  Rh ) k k!μ k H + C  k=C−C h +1 (  R +  Rh ) C−C h  k− ( C−C h ) Rh k!μ H k  −1 P j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (  R +  Rh ) j j!μ j H P 0 , for j = 1, 2, ,C −C h (  R +  Rh ) C−C h  j− ( C−C h ) Rh j!μ j H P 0 , for j = C −C h + 1, ,C (11.14) The probability of blocking a new call is P B =  C j=C−C h P j and the probability of handoff attempt failure P fh is the probability that the state number of the base station is equal to C. Thus P fh = P c . 11.2.3 Channel reservation with queueing – CRQ handoffs When a mobile moves away from the base station, the received power generally decreases. When the received power gets lower than a handoff threshold level, the handoff procedure JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 334 MOBILITY MANAGEMENT Delayed Blocked Handoff attempts New call originators Queue of Delayed Handoff Attemps Channel use record Delayed Blocked Handoff attempts New call originators Queue of delayed handoff attemps Channel use record Calls in progressForced terminations Figure 11.22 Call flow diagram for channel reservation with queueing-CRQ handoffs. is initiated. The handoff area is defined as the area in which the average received power level from the base station of a mobile receiver is between the handoff threshold level (upper bound ) and the receiver threshold level (lower bound ). If the handoff attempt finds all channels in the target cell occupied, we consider that it can be queued. If any channel is released while the mobile is in the handoff area, the next queued handoff attempt is accomplished successfully. If the received power level from the source cell’s base station falls below the receiver threshold level prior to the mobile being assigned a channel in the target cell, the call is forced into termination. When a channel is released in the cell, it is assigned to the next handoff call attempt waiting in the queue (if any). If more than one handoff call attempt is in the queue, the first-come-first-served queuing discipline is used. The prioritized queueing is also possible where the fast moving (fast signal level losing) users may have higher priority. We assume that thequeuesizeatthebasestationisunlimited. Figure 11.22 shows a schematic representation of the flow of call attempts through a base station. The time for which a mobile is in the handoff area depends on system parameters such as the speed and direction of mobile travel and the cell size. We call it the dwell time of a mobile in the handoff area T Q . For simplicity of analysis, we assume that this dwell time is exponentially distributed with mean ¯ T Q (1/μ H ). We define E j as the state of the base station when j is the sum of the number of channels being used in the cell and the number of handoff call attempts in the queue. For those states whose state number j is less than equal to C, the state transition relation is the same as for the CR scheme. Let X be the elapsed time from the instant a handoff attempt joins the queue to the first instant that a channel is released in the fully occupied target cell. For state numbers less than C, X is equal to zero. Otherwise, X is the minimum remaining holding time of those calls in progress in the fully occupied target cell. When a handoff attempt joins the queue for a given target cell, other handoff attempts may already be in the queue (each is associated with a particular mobile). When any of these first joined the queue, the time that it could remain on the queue without succeeding is denoted by T Q (according to our previous definition). Let T i be the remaining dwell time for that attempt which is in the ith queue position when another handoff attempt JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF 335 0 E 1 E C-Ch E C E C+1 E μH 2μ H (C-C H )μ Η (C-C H +1)μ Η E 0 E 1 E C-C h E C+1 Λ R + Λ Rh Λ R + Λ Rh Λ R + Λ Rh Λ Rh Λ Rh Λ Rh Λ Rh Cμ H Cμ H +μ Q Cμ H +2μ H E C Figure 11.23 State-transition diagram for CRQ priority scheme. joins the queue. Under the memoryless assumptions here, the distributions of all T i and T Q are identical. Let N ( t ) be the state number of the system at time t. From the description of this scheme and the properties of the exponential distribution it follows that P r {N(t + h) = C +k − 1|N(t) = C +k} = P r {X ≤ h or T 1 ≤ h or T k ≤ h} = 1 − P r {X > h and T 1 > h or T k > h} (11.15) = 1 − P r { X > h } P r { T 1 > h } P r { T k > h } = 1 − e − ( C μ H +kμ Q ) h since the random variables X, T 1 , T 2 , ,T k are independent. From Equation (11.15) we see that it follows the birth-and-death process and the resulting state transition diagram is as shown in Figure 11.23. As before, the probability distribution {P j } is easily found to be P 0 = ⎡ ⎢ ⎢ ⎢ ⎣ C−C h  k=0 (  R +  Rh ) k k!μ H k + C  k=C−C h +1 (  R +  Rh ) C−C h  k− ( C−C h ) Rh k!μ k H + ∞  k=C+1 (  R +  Rh ) C−C h  k− ( C−C h ) Rh C!μ C H k−C  i=1  Cμ H +iμ Q  ⎤ ⎥ ⎥ ⎥ ⎦ −1 P j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (  R +  Rh ) j j!μ j H P 0 , for 1 ≤ j ≤ C −C h (  R +  Rh ) C−C h  j− ( C−C h ) Rh j!μ j H P 0 , for C − C h + 1 ≤ j ≤ C (  R +  Rh ) ( C−C h )  j− ( C−C h ) Rh C!μ C H j−C  i=1  Cμ H +iμ Q  P 0 , for j ≥ C +1 (11.16) The probability of blocking is P B =  ∞ j=C−C h P j . A given handoff attempt that joins the queue will be successful if both of the following events occur before the mobile moves out JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 336 MOBILITY MANAGEMENT of the handoff area: (1) All of the attempts that joined the queue earlier than the given attempt have been disposed. (2) A channel becomes available when the given attempt is at the front of the queue. Thus the probability of a handoff attempt failure can be calculated as the average fraction of handoff attempts whose mobiles leave the handoff area prior to their coming into the queue front position and getting a channel. Noting that arrivals that find k attempts in queue enter position k +1, this can be expressed as P fh  ∞  k=0 P C+k P fh | k (11.17) whereP fh | k = P r { attempt fails given it enters the queue in position k − 1}. Since handoff success for those attempts which enter the queue in position k +1 requires coming to the head of the queue and getting a channel, under the memoryless conditions assumed in this development, we have  1 − P fh | k  =  k  i=1 P ( i | i + 1 )  P r  get channel in first position} (11.18) whereP ( i | i + 1 ) is the probability that an attempt in position i + 1 moves to position i before its mobile leaves the handoff area. There are two possible outcomes for an attempt in position i +1. It will either be cleared from the system or will advance in queue to the next (lower) position. It will advance if the remaining dwell time of its mobile exceeds either: (1) at least one of the remaining dwell times T j , j = 1, 2, ,i, for any attempt ahead of it in the queue; or (2) the minimum remaining holding time X of those calls in progress in the target cell. Thus 1 − P(i | i + 1) = P r {T i+1 ≤ X, T i+1 ≤ T j , j = 1, 2, ,i} i = 1, 2, (11.19) 1 − P(i | i + 1) = P r {T i+1 ≤ X, T i+1 ≤ T 1 , ,T i+1 ≤ T i } = P r {T i+1 ≤ min(X, T 1 , T 2 , ,T i )} (11.19a) = P r {T i+1 ≤ Y i } i = 1, 2, where Y i ≡ min(X, T 1 , T 2 , ,T i ). Sincethemobiles move independentlyofeachother and of the channel holding times, the random variables, X, T j , ( j = 1, 2, ,i) are statistically independent. Therefore, the cumulative distribution of Y i in Equation (11.19) can be written as F Y i (τ ) = 1 −{1 − F X (τ )}{1 − F T 1 (τ )} {1 − F T i (τ )} JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF 337 Because of the exponentially distributed variables, this gives F Y i (τ ) = 1 −e −Cμ H τ e −μ Q τ e −μ Q τ = 1 −e −(Cμ H +iμ Q )τ and Equation (11.19) becomes 1 − P(i | i + 1) = P r {T i+1 ≤ Y i }= ∞  0 {1 − F Y i (τ )}f T i+1 (τ )dτ (11.20) = ∞  0 e −(Cμ H +iμ Q )τ μ Q e −μ Q τ dτ = μ Q Cμ H + (i + 1)μ Q , i = 1, 2, The handoff attempt at the head of the queue will get a channel (succeed) if its remaining dwell time T 1 exceeds X. Thus P r {get channel in front position}=P r {T 1 > X}and P r { does not get channel in front position } = P r { T 1 ≤ X } (11.21) = ∞  0 e −C μHτ μ Q e −μ Q τ dτ = μ Q Cμ H + μ Q The probability Equation (11.21) corresponds to letting i = 0 in Equation (11.20) Then from Equation (11.18) we have 1 − P fh | k =  k  i=1 P ( i | i + 1 )  P r { get channel in first position} = Cμ H + μ Q Cμ H + 2μ Q Cμ H + 2μ Q Cμ H + 3μ Q ··· Cμ H + kμ Q Cμ H + ( k +1 ) μ Q Cμ H Cμ H + μ Q (11.22) = Cμ H Cμ H + ( k +1 ) μ Q and P fh | k = ( k +1 ) μ Q Cμ H + ( k +1 ) μ Q (11.23) The above equations form a set of simultaneous nonlinear equations which can be solved for system variables when parameters are given. Beginning with an initial guess for the un- knowns, the equations are solved numerically using the method of successive substitutions. A call which is not blocked will be eventually forced into termination if it succeeds in each of the first ( l −1 ) handoff attempts which it requires but fails on the lth. Therefore, P F = ∞  l=1 P fh  P n ( 1 − P fh ) l−1 P l−1 H  = P fh P N 1 − P H ( 1 − P fh ) (11.24) where P N and P H are the probabilities of handoff demand of new and handoff calls, as defined previously. Let P nc denote the fraction of new call attempts that will not be com- pleted because of either blocking or unsuccessful handoff. This is also an important system JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 338 MOBILITY MANAGEMENT performance measure. This probability P nc can be expressed as P nc = P B + P F ( 1 − P B ) = P B + P fh P N ( 1 − P B ) 1 − P H ( 1 − P fh ) (11.25) where the first and second terms represent the effects of blocking and handoff attempt failure, respectively. In Equation (11.25) we can guess roughly that, when cell size is large, the probabilities of cell crossing P N and P H will be small and the second term of Equation (11.25) (i.e. the effect of cell crossing) will be much smaller than the first term (i.e. effect of blocking). However, when the cell size is decreased, P N and P H will increase. The noncompleted call probability P nc can be considered as a unified measure of both blocking and forced termination effects. Another interesting measure of system performance is the weighted sum of P B and P F CF = (1 −α)P B + α P F (11.26) where α is in the interval [ ( 0, 1 ) ] and indicates the relative importance of the blocking and forced termination effects. For some applications P F may be more important than P B from the user’s point of view, and the relative cost α can be assigned using the system designer’s judgment. 11.2.4 Performance examples Forthecalculations, theaverage messagedurationwas takenas ¯ T M =120sandthe maximum speed of a mobile of V max = 60 miles/h was used. The probabilities P B and P F as functions of (new) call origination rate per unit area  a can be seen in Figure 11.24, with cell radius R being a parameter. A total of 20 channels per cell ( C = 20 ) and one channel per cell for handoff priority (C h = 1) was assumed. The CRQ scheme was used for this figure, and the mean dwell time for a handoff attempt ¯ T Q was assumed to be ¯ T H /10. As can be seen, P F is much smaller than P B and the difference between them decreases as cell size decreases. As expected, for larger R the effect of handoff attempts and forced terminations on system performance is smaller. Call origination rate density (calls per s/square mile) 10 -4 10 -4 10 -3 10 -3 10 -2 10 -2 10 -1 10 -1 10 0 10 0 R =16 R =4 R =4 R =16 R =1 R =1 Probability Priority scheme II 20 channels/cell 1 handoff channel/cell blocking forced termination Figure 11.24 Blocking and forced termination probabilities for CRQ priority scheme. JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF 339 10 -3 10 -4 10 -3 10 -2 10 -1 10 0 10 -2 10 -1 4 2 C h = 0 Probability Blocking Forced termination C h = 4 2 0 Call ori g ination rate density (calls per s/square mile) Figure 11.25 Blocking and forced termination probabilities for CRQ systems with 20 chan- nels/cell, R = 2 miles. Call origination rate density (calls per s/square mile) 10 -3 10 -3 10 -2 10 -2 10 -1 10 -1 10 0 CR P B CRQ 10 0 Probability P F Figure 11.26 Blocking and forced terminations for priority CR and CRQ schemes (20 channels/cell, one handoff channel/cell, R = 2 miles). Figure 11.25 shows P B and P F as functions of  a . As the effects of increasing priority given to handoff calls over new calls by increasing C h , P F decreases by orders of magnitude with only small to moderate increase in P B this exchange is important because (as was mentioned previously) forced terminations are usually considered much less desirable than blocked calls. Blocking and forced termination probabilities for the two priority schemes are shown in Figure 11.26 as functions of call origination rate density  a . The forced termination probability P F is smaller for th CRQ scheme, but almost no difference exists in blocking probability P B . We get this superiority of the CRQ priority scheme by queuing the delayed handoff attempts for the dwell time of the mobile in the handoff area. JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 340 MOBILITY MANAGEMENT 11.3 CELL RESIDING TIME DISTRIBUTION In this section we discuss the probability distributions of the residing times T n and T h . The random variable T n is defined as the time (duration) that a mobile resides in the cell in which its call originated. Also T h is defined as the time a mobile resides in a cell to which its call is handed off. To simplify analysis we approximate the hexagonal cell shape as a circle. For a hexagonal cell having radius R, the approximating circle with the same area has a radius, R eq , which is given by R eq = √ (3 √ 3/2π) R ≈ 0.91R and illustrated in Figure 11.27. The base station is assumed to be at the center of a cell and is indicated by a letter B in the figure. The location of a mobile in a cell, which is indicated by a letter A in the figure, is represented by its distance r and direction φ from the base station as shown. To find the distributions of T n and T h , we assume that the mobiles are spread evenly over the area of the cell. Then r and φ are random variables with PDFs f r (r) = ⎧ ⎨ ⎩ 2r R 2 eq , 0 ≤ r ≤ R eq 0, elsewhere , f φ (φ) = ⎧ ⎨ ⎩ 1 2π , 0 ≤ φ ≤ 2π 0, elsewhere (11.27) Next it is assumed that a mobile travels in any direction with equal probability and its direction remains constant during its travel in the cell. If we define the direction of mobile travel by the angle θ (with respect to a vector from the base station to the mobile), as shown in the figure, the distance Z from the mobile to the boundary of approximating circle is Z = √ [R 2 eq − (r sin θ) 2 ] −r cos θ. Because φ is evenly distributed in a circle, Z is independent of φ and from the symmetry we can consider the random variable θ is in interval [ 0,π ] with PDF f θ (θ) = ⎧ ⎨ ⎩ 1 π , 0 ≤ θ ≤ π 0, elsewhere (11.28) y = rsin q x = rcos q R B Z q R eq φ C A r φ Figure 11.27 Illustration of distance from point A in cell (where call is originated), to point C on cell boundary (where mobile exits from cell). [...]... mobile communication networks, ACM/Baltzer Wireless Networks J., vol 3, no 2, 1997, pp 159 –167 [50 ] C Rose and R Yates, Location uncertainty in mobile networks: a theoretical framework, IEEE Commun Mag., vol 35, 1997, pp 94–101 [51 ] C Rose, Minimizing the averagecost of paging and reg-istration: a timer-based method, ACM-Baltzer J Wireless Networks, vol 2, no 2, 1996, pp 109–116 [52 ] C Rose and R Yates,... 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In a micro- and picocellular. P B ) 1 − P H ( 1 − P fh ) (11. 25) where the first and second terms represent the effects of blocking and handoff attempt failure, respectively. In Equation (11. 25) we can guess roughly that, when. time in a cell as suggested in References [ 75, 76]. The goodness-of-fit JWBK083-11 JWBK083-Glisic March 8, 2006 20:12 Char Count= 0 344 MOBILITY MANAGEMENT 50 60 70 80 90 100 110 120 130 02468101214 Cell